Introduction to Particle Physics. Sreerup Raychaudhuri TIFR. Lecture 5. Weak Interactions

Transcription

1 Introduction to Particle Physics Sreerup Raychaudhuri TIFR Lecture 5 Weak Interactions

2 Pauli s neutrino hypothesis 1

3 2 Fermi s theory of beta decay 1 1 0n 1 p + e νe p + n The decay must take place through weak interactions (τ = 887 s). Can we write down an interaction vertex? First attempted by Fermi (1934) νe

4 3 Denote the Dirac fields: Ψ n = n, Ψ p = p, Ψ e = e and Ψ νe = νe Fermi s first attempt: try a four-fermion vertex n p + ig F νe Weak interaction Hamiltonian: H I = G F 2 pn e ν e dimension of G F is M 2 : Fermi coupling constant Simplest possible form of a four-fermion coupling

5 4 With this interaction, the probability for the transition, in the restframe of the neutron, comes out to be M 2 4G F 2 M n M p E e 2 1 cos θ eν i.e. the electron and the antineutrino should tend to come out back-toback... Actual experiment showed that, instead, the electron and the antineutrino tended to come out in the same direction! More as if we have M cos θ eν Fermi s second attempt: try a vertex modelled on e.m. interactions, H I = G F 2 pγμ n e γ μ ν e = G F 2 J μ had J μ lep Current-current form of the weak interaction

6 5 With this interaction, the probability for the transition, in the restframe of the neutron, comes out to be M 2 8G F 2 M n M p E e cos θ eν This fits the experimental data much better... Total decay width (rough estimate): Γ β G F π s where = M n M p From this we can estimate G F GeV 2 Given the crudeness of the approximation, this is not a bad estimate... Current value: G F = GeV 2

7 6 More important: 2 dγ β G F de e 2π 3 E e 3 E e 1 E e 3 2 dγ β G F de e 2π 3 E e 1 E e 3 dγ β de e Kurie plot for tritium decay E e (kev) Fermi s theory is spectacularly successful in explaining beta energy spectrum

8 7 In 1937, the muon was discovered... it decays to electron... Decay must be through weak interactions (tracks are seen)... μ + ν e + ν μ Fermi s guess: universality of weak interactions μ ν μ ig F νe

9 8 Use this to calculate the muon lifetime: τ μ 192 π3 G F 2 M μ s Spectacular agreement with the experimental value s Vindicates Fermi s hypothesis about universality of weak interactions... today we have many more proofs... Interestingly, Fermi could have written several forms of the interaction, e.g. H I = G F 2 pγμ γ 5 n e γ μ γ 5 ν e or H I = G F 2 pσμν n e σ μν ν e The choice of the vector-vector form turned out to be a stroke of genius, for that is exactly what we predict in the gauge theory of weak interactions which is what the Fermi theory ultimately leads to...

10 9 Weak scattering processes: the unitarity problem If we have a vertex μ ν μ ig F νe as Fermi postulated, then, by universality, we should also have ν e ig F νe

11 10 and it should be possible to have a scattering process Cross-section: + ν e + ν e σ G F 2 π s 1 M 2 e s where s = p e + p 2 2 νe = E cm. Clearly, as s, σ... unitarity violation Perhaps this arises because we took only the LO diagram...?... inclusion of higher orders may soften the growth with energy......but this leads to a new problem: renormalisability

12 11 Consider the simplest one-loop contribution to ν e ν e : p a k p 1 ig F ig F ν e p b k + p a + p b ν e p 2 ν e The effective coupling due to this would be ig F 2 ig F 2 + ig F 2 2 d 4 k 1 1 2π 4 k M e k + p a + p b Since k is integrated over all values, the dominant contribution will come from k, i.e.

13 12 ig F 2 ig F 2 + ig F 2 2 = ig F 2 + ig F = ig F 2 + ig F 2 k dk 16π π 2 k 3 dk 2π 4 2π 2 k 3 dk 2π 4 This extra contribution is quadratically divergent, i.e. if we put a momentum cutoff k Λ then, ig F 2 ig F 2 + ig F 2 k dk 16π2 = ig F 2 + ig F 2 32π 2 0 Λ2 Λ If the NLO contribution LO contribution, perturbation theory fails... 1 kk 1 k 2

14 13 Such problems arise in QED as well for e, but there the divergences are logarithmic, i.e. proportional to log Λ. Moreover, in every order (NLO, NNLO, NNNLO,...) we always get a similar logarithmic divergence. These can be summed up, and the result absorbed into the definition of e -- this process is called renormalisation In the Fermi theory, however, higher and higher powers of Λ 2 keep coming with higher and higher orders, and there is no scope for renormalisation... Does this mean that the Fermi theory is wrong?

15 14 Such problems arise in QED as well for e, but there the divergences are logarithmic, i.e. proportional to log Λ. Moreover, in every order (NLO, NNLO, NNNLO,...) we always get a similar logarithmic divergence. These can be summed up, and the result absorbed into the definition of e -- this process is called renormalisation In the Fermi theory, however, higher and higher powers of Λ 2 keep coming with higher and higher orders, and there is no scope for renormalisation... Does this mean that the Fermi theory is wrong? Correspondence Principle: every new theory should reduce to the old theory in the range of parameters where that theory was successful Fermi theory must be a low-energy effective theory...

16 15 Intermediate Vector Bosons (IVB): Schwinger (1953): if renormalisation is possible in QED, can we make it possible in weak interactions by copying the same form? Consider the following process in QED: + e + μ + μ + p a ie k photon p 1 ie μ Taking Fermi s idea a step further... e + p b p 2 μ + im = v p b ieγ μ u p a ig μν k 2 u p 1 ieγ ν v p 2 = ie2 k 2 v p b γμ u p a u p 1 γ μ v p 2 G F 2 e2 k 2

17 16 Intermediate Vector Bosons (IVB): Schwinger (1953): if renormalisation is possible in QED, can we make it possible in weak interactions by copying the same form? Consider the following weak process: + νe μ + νμ p a ig k W p 1 ig μ Taking Fermi s idea a step further... νe p b p 2 νμ im = v p b igγ μ u p a ig μν k 2 u p 1 igγ ν v p 2 = ig2 k 2 v p b γμ u p a u p 1 γ μ v p 2 G F 2 = g2 k 2

18 17 Intermediate Vector Bosons (IVB): Schwinger (1953): if renormalisation is possible in QED, can we make it possible in weak interactions by copying the same form? Consider the following weak process: + νe μ + νμ W boson vertex p a ig k W p 1 ig μ Taking Fermi s idea a step further... νe p b p 2 νμ im = v p b igγ μ u p a ig μν k 2 u p 1 igγ ν v p 2 = ig2 k 2 v p b γμ u p a u p 1 γ μ v p 2 G F 2 = g2 k 2

19 18 Intermediate Vector Bosons (IVB): Schwinger (1953): if renormalisation is possible in QED, can we make it possible in weak interactions by copying the same form? Consider the following weak process: + νe μ + νμ W propagator p a ig k W p 1 ig μ Taking Fermi s idea a step further... νe p b p 2 νμ im = v p b igγ μ u p a ig μν k 2 u p 1 igγ ν v p 2 = ig2 k 2 v p b γμ u p a u p 1 γ μ v p 2 G F 2 = g2 k 2

20 19 Objection: The Fermi coupling constant does not show significant variation with energy as k 2 0 Schwinger s solution: make the W boson massive W: ig μν k 2 im = v p b igγ μ u p a ig μν +k μ k ν /M W 2 ig 2 μν + k μ k ν /M W k 2 2 M W k 2 M W 2 u p 1 igγ ν v p 2 G F 2 = g2 k 2 = v p b igγ μ u p a ig μν k 2 M W 2 u p 1 igγ ν v p 2 = ig2 k 2 M W 2 v p b γ μ u p a u p 1 γ μ v p 2 G F 2 = g 2 k 2 M W 2 In the low energy limit, k 2 0 we get: G F 2 = g2 M W 2 constant!!

21 20 Q. How does this help? p a k p 1 ig F ig F ν e p b k + ν e p 2 ν e ig k ig W k + W k + ν e ig k + ig ν e ν e

22 21 Rewrite the loop integral... d 4 k 2π 4 1 k 1 k + d 4 k 2π 4 1 k 1 1 k 2 2 M W k + 1 k 2 M W 2 k 3 dk As it would be in QED... dk k k 2 k 4 finite!

23 22 Rewrite the loop integral... d 4 k 2π 4 1 k 1 k + d 4 k 2π 4 1 k 1 1 k 2 2 M W k + 1 k 2 M W 2 k 3 dk As it would be in QED... dk k k 2 k 4 finite! But we have cheated... ig μν k 2 ig 2 μν +k μ k ν /M W k 2 M2 W d 4 k 2π 4 1 k k μ k ν k 2 M W 2 1 k + k μ k ν k 2 M W 2 k3 dk 1 k 4 k 2 k 4 kdk Only way to make IVB work is to get rid of the k μ k ν /M W 2 term...

24 23 What does the propagator couple to? p a p 1 j in μ ig k W ig ν j ou t p b p 2 M j μ in ig ig 2 μν + k μ k ν /M W ν k 2 2 ig j M ou t W

25 24 What does the propagator couple to? p a p 1 j in μ ig k W ig ν j ou t p b p 2 M j μ in ig ig 2 μν + k μ k ν /M W ν k 2 2 ig j M ou t W The offending term will go away if j μ ν in k μ = 0 and/or k ν j ou t = 0 To have conserved currents, there must be a gauge symmetry...

26 25 But this cannot be a U(1) gauge symmetry, like QED Why not? Because the W boson is charged, i.e. there are two W bosons W μ + = 1 2 W iw 2 + W μ = 1 2 W 1 + iw 2 + i.e. the group of gauge symmetries must have at least two generators In fact, if we have a four-fermion theory with the vertex ν e ig F there is nothing, in principle, to prevent a process like νe + e + ν e + νe

27 26 How will this look in the IVB theory? p a p 1 ν e ig k W ig e + p b p 2 νe So, perhaps we have a neutral W boson also a W μ 0 This W μ 0 cannot be the photon because it couples to neutrinos... i.e. the group of gauge symmetries must have three generators after U(1), the next unitary group is SU(2), which has 3 generators...

28 27 Parity violation : the θ τ puzzle Consider the Fermi form of the current-current interaction: Under parity: H I = G F 2 pγμ n (e γ μ ν e ) pγ μ n pγ μ n (e γ μ ν e ) (e γ μ ν e ) i.e. parity is conserved in the Fermi theory Before the 1950s, it was thought that parity is as sacred as energy, momentum and angular momentum... But it was known that some particle are pseudoscalars, This led to the famous θ τ puzzle e.g. pions and Kaons have intrinsic parity P = 1

29 28 Through the early 1950s, cosmic ray experiments showed the existence of two degenerate particles θ and τ, each with mass around 483 MeV and lifetime around 12 ns. However, it was seen that θ + π + + π 0 τ + π + + π + π + indicating that P θ = +1 and P τ = 1. Note that the phase space for these decays is very different: M(θ + ) M π + M(π 0 ) = = 218 MeV M(τ + ) M π + M(π ) M(π + ) = = 73 MeV Since the lifetimes are identical, the strength of weak interactions must be different for these different decays universality is violated

30 29 Yang & Lee (1956) Maybe parity is not conserved in weak interactions, i.e. θ + π + + π 0 is really K + π + + π 0 parity-violating channel τ + π + + π + π + is really K + π + + π + π + parity-conserving channel They also showed that none of the earlier experiments had really tested intrinsic parity violation... Suggested that if the mean value of a party-odd variable, e.g. S. p could be found to be nonzero, this would be a smoking gun signal for parity violation Experiment was actually performed by Wu et al (1957)...

31 30 Maximal parity violation: In 1956, Marshak & Sudarshan, and separately, Feynman & Gell-Mann, assumed the parity-violating weak interactions to be of the form H I = G F 4 2 pγμ 1 λγ 5 n. e γ μ 1 λγ 5 ν e Parity is conserved when λ = 0 (V current), λ (A current) Parity is maximally violated when λ = 1 (V-A currents) Parity is partially violated for other values of λ... Rewrite the leptonic current as μ J lep = e γ μ 1 λγ 5 ν e = 1 + λ e L γ μ ν el + 1 λ e R γ μ ν er

32 31 If we can measure the chirality of neutrinos emitted in beta decay, then we should have and hence P ν el = P ν er = 1 + λ λ λ 2 1 λ λ λ 2 P ν er P ν = 1 λ el 1 + λ Goldhaber et al did an experiment in 1957 with the electron capture process Eu Sm 62 + ν e and found that the neutrino is always left-chiral... It follows that λ =

33 32 Thus the weak interactions do have the form V A and the W boson vertex for electrons is of the form H I = g 2 2 e γα 1 γ 5 ν e W g α ν eγ α + 1 γ 5 ew α We will have similar interactions for the muon H I = g 2 2 μ γα 1 γ 5 ν μ W g α ν μ γ α + 1 γ 5 μw α and for the nucleons and for the quarks H I = g 2 2 p γα 1 γ 5 nw g α nγα + 1 γ 5 pw α H I g 2 2 d γ α 1 γ 5 uw g α uγα + 1 γ 5 dw α